According to classical thin airfoil theory for an arbitrary cambered thin airfoil in incompressible, inviscid flow, the pitch moment coefficient about the aerodynamic center $C_{m,ac}$ is independent of angle of attack $\alpha$. The theoretical chordwise location of this aerodynamic center $(x_{ac}/c)$ from the leading edge is:

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles From thin airfoil theory, the Fourier expansion of the circulation distribution yields: $$C_l = 2\pi \left(\alpha + \frac{A_1}{2} - A_0\right) = 2\pi(\alpha - \alpha_{L=0})$$ The moment coefficient about the leading edge is: $$C_{m,le} = -\frac{C_l}{4} + \frac{\pi}{4}(A_1 - A_2)$$ ### 2. Analytical Formulation The pitch moment coefficient about an arbitrary point $x$ from the leading edge is: $$C_{m,x} = C_{m,le} + \frac{x}{c} C_l = \left(\frac{x}{c} - \frac{1}{4}\right) C_l + \frac{\pi}{4}(A_1 - A_2)$$ By definition, the aerodynamic center is the point where $\frac{\partial C_m}{\partial C_l} = 0$: $$\frac{\partial C_{m,x}}{\partial C_l} = \frac{x}{c} - \frac{1}{4} = 0 \implies \frac{x_{ac}}{c} = \frac{1}{4} = 0.25$$ ### 3. Step-by-Step Calculation Regardless of the camber line shape (i.e., for any Fourier coefficients $A_0, A_1, A_2, \dots$), the aerodynamic center in subsonic thin airfoil theory is strictly at the quarter-chord point: $$\frac{x_{ac}}{c} = 0.25$$ ### 4. Distractor Forensics - At supersonic speeds, linearized supersonic airfoil theory places the aerodynamic center at the mid-chord ($x_{ac}/c = 0.50$). Candidates frequently confuse subsonic (0.25) with supersonic (0.50). ### 5. Exam Trap & Standard Reference - **Trap Alert:** Incompressible thin airfoil theory always yields $x_{ac}/c = 0.25$; supersonic linearized theory yields $x_{ac}/c = 0.50$. - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 4 (Incompressible Flow over Airfoils).

← Prev Question Next Question →